3.2 – Use Parallel Lines and Transversals

Identify the angle pairs.

5

1 2

3

7 8

4 6

1.) < 1 𝑎𝑛𝑑 < 6

Alt. Exterior Angles

2.) < 4 𝑎𝑛𝑑 < 7

Alt. Interior Angles

3.) < 3 𝑎𝑛𝑑 < 4

Corresponding Angles

4.) < 2 𝑎𝑛𝑑 < 7

Consec. Interior Angles

Objective: Students will be able to use parallel lines and specific angle pairs to find angle measures.

AgendaPostulates/Practice

Theorems/Practice

Proving Theorems

Postulate 15 – Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

1

5

𝒎

𝒏

𝒕

𝒎 ∥ 𝒏

< 𝟏 ≅ < 𝟓

Theorem 3.1– Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

7

3

𝒎

𝒏

𝒕

𝒎 ∥ 𝒏

< 𝟑 ≅ < 𝟕

Use the diagram to find 𝑚 < 3 and 𝑚 < 5.

𝒎 < 𝟓 = 𝟏𝟒𝟎° (Alt. Int. <‘s)𝒎 < 𝟑 = 𝟏𝟒𝟎° (Corr. <‘s)

Theorem 3.2– Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

4

6

𝒎

𝒏

𝒕

𝒎 ∥ 𝒏

< 𝟒 ≅ < 𝟔

Use the diagram to find 𝑚 < 5 and 𝑚 < 6.

𝒎 < 𝟓 = 𝟖𝟕° (Alt. Int. <‘s)𝒎 < 𝟔 = 𝟗𝟑° (Alt. Ext.<‘s)

Theorem 3.3– Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

2

8

𝒎

𝒏

𝒕 𝒎 ∥ 𝒏

< 𝟐 𝐚𝐧𝐝 < 𝟖𝐚𝐫𝐞 𝐬𝐮𝐩𝐩𝐥𝐞𝐦𝐞𝐧𝐭𝐚𝐫𝐲

𝒎 < 𝟐+𝒎 < 𝟖 = 𝟏𝟖𝟎°

Use the diagram to find 𝑚 < 3,𝑚 < 4, and 𝑚 < 7.

𝒎 < 𝟑 = 𝟏𝟑𝟓° (Alt. Ext. <‘s)𝒎 < 𝟒 = 𝟏𝟑𝟓° (Consec. Int.<‘s)𝒎 < 𝟕 = 𝟒𝟓° (Corr. <‘s)

Use the given diagram to find the value of x.

Equation:

115 + 𝑥 + 5 = 180

120 + 𝑥 = 180

𝒙 = 𝟔𝟎

115°4

(𝑥 + 5)°

Given the diagram, give a statement that can be made using the following postulate/theorem.

1.) Corresponding Angles Theorem

< 1 ≅< 2

2.) Alternate Exterior Angles Theorem

< 3 ≅< 8

3.) Consecutive Interior Angles Theorem

𝑚 < 5 +𝑚 < 4 = 180°

4.) Alternate Interior Angles Theorem

< 4 ≅< 7

Given: 𝑚 ∥ 𝑛

Prove: < 2 ≅ < 3

Statements Reasons

1. Given1.𝑚 ∥ 𝑛

2. < 1 ≅ < 2

3. < 1 ≅ < 3

2. Vertical Angle Congruence Theorem

3. Corresponding Angles Postulate

4. < 2 ≅ < 3 4. Transitive Property

Given: 𝑚 ∥ 𝑛

Prove: < 1 ≅ < 2

Statements Reasons

1. Given1.𝑚 ∥ 𝑛

2. < 2 ≅ < 3

3. < 1 ≅ < 3

2. Vertical Angle Congruence Theorem

3. Corresponding Angles Postulate

4. < 1 ≅ < 2 4. Transitive Property

Given: 𝑚 ∥ 𝑛

Prove: < 4 and < 5 are supplementary

Statements Reasons 1. Given1.𝑚 ∥ 𝑛

2. < 4 ≅ < 6

4. < 5 and < 6 are supplementary

2. Alternate Interior Angles Theorem

4. Linear Pair Postulate

5. 𝑚 < 5 +𝑚 < 6 = 180° 5. Def. of Supplementary Angles

3. 𝑚 < 4 = 𝑚 < 6 3. Def. of Congruent Angles

6. 𝑚 < 5 +𝑚 < 4 = 180° 6. Substitution Property

7.< 4 and < 5 are supplementary 7. Def. of Supplementary Angles